ETNA
Electronic Transactions on Numerical Analysis.
Volume 29, pp. 70-80, 2008.
Copyright  2008, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
HIERARCHICAL GRID COARSENING FOR THE SOLUTION OF THE POISSON
EQUATION IN FREE SPACE
MATTHIAS BOLTEN
Abstract. In many applications the solution of PDEs in infinite domains with vanishing boundary conditions
at infinity is of interest. If the Green’s function of the particular PDE is known, the solution can easily be obtained
by folding it with the right hand side in a finite subvolume. Unfortunately this requires
operations. Washio
and Oosterlee presented an algorithm that rather than that uses hierarchically coarsened grids in order to solve the
problem (Numer. Math. (2000) 86: 539–563). They use infinitely many grid levels for the error analysis. In this
paper we present an extension of their work. Instead of continuing the refinement process up to infinitely many
grid levels, we stop the refinement process at an arbitrary level and impose the Dirichlet boundary conditions of the
original problem there. The error analysis shows that the proposed method still is of order , as the original method
with infinitely many refinements.
Key words. the Poisson equation, free boundary problems for PDE, multigrid method
AMS subject classifications. 35J05, 35R35, 65N55
1. Introduction. Applications such as flow simulations around aircrafts (c.f. [4]) and
molecular dynamics simulations (see [8]) require so-called free or open boundary conditions.
Consider for example the Poisson equation with vanishing boundary conditions at infinity
!
#"%$&(')' *')'+"-,.
/0
where supp
is a bounded subset of , as needed in molecular dynamics simulations to
calculate the interaction between charges due to the electrostatic potential.
21 be a discretization of the Laplace operator
This problem has a discrete analog. Let
on a infinite regular grid with mesh-width 3 . Then the infinitely large system of equations
given by
1 456
798:;'< 3= = >
?@
A"%$BC'D' *'D'@"E,
is the discretization of the continous problem.
There are different approaches to solve the discrete problem. We just want to mention a
few of them:
1. Solution in a finite subvolume by explicitly imposing boundary values.
2. Solving the problem on a hierarchically coarsened infinitely large grid.
3. Solution of the problem on a hierarchically coarsened finite grid by imposing boundary values.
The first approach was chosen by Buneman in [3] to solve a two-dimensional potential prob-1
lem. He derived a recursive formula for the discrete Green’s function, i.e., a function F
which satisfies
L
G1 F B1 6HJIC.K:1B6HJI
Received November 3, 2006. Accepted for publication June 28, 2007. Published online on February 7, 2008.
Recommended by A. Frommer.
Jülich Supercomputing Centre, Research Centre Jülich, D-52425 Jülich, Germany
(m.bolten@fz-juelich.de).
70
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HIERARCHICAL GRID COARSENING
K<1
71
K
is the discrete Dirac- . This yielded an exact solution of the discrete problem,M
i.e.
where
1.
a solution of the continuous problem that is only disturbed by the discretization error of
Later
in [4], Burkhart derived an asymptotic expansion of the discrete Green’s function in
thatonallows
the same proceedure for the three-dimensional case. This asymptotic expansion is given by
^]_ 3 S .
R N
1
#
O
N
T XYX<X[Z P@Q 'D' *D' ' SGT 'D' *U ')' S T ')' *UWV ')' S .
F
V
where
U@\
Unfortunately these proceedures share the high computational costs of imposing the
Dirichlet boundary conditions explicitly by using the recursive formula or the asymptotic
expansion. Consider a cube with `7acb grid points in each direction. In order to compute the
values at all boundary points, the formula or the expansion has to be evaluated
S
d `fe<`^hWg i d `khj
times. So the overall complexity
with exponent l+mon , which is unsatisfying
]_ ` is ,polynomial
when one uses an optimal, i.e.
Poisson solver.
When dealing with particle systems, this problem can be avoided by using a multipole
expansion in order to evaluate the potential on the boundary, as proposed by Sutmann and
Steffen in [8].
The second approach was introduced by Washio and Oosterlee in [9]. They introduced
a hierarchical grid coarsening, which uses the idea of locally refined grids, in order to reduce
the computational cost outside of the subdomain of interest. They do this by extending the
S at the same
S time. They
original grid hierarchically by a factor p , while coarsening the grid
have shown that the result of this process is accurate to order 3 , if prqts b . For their
numerical experiments, they have chosen to stop the process after a number of steps and
impose Dirichlet zero boundary conditions there.
In this paper we have extended their work imposing Dirichlet boundary conditions of
the continuous problem at the level at which we stop the refinement. Since the number of
boundary points is dramatically reduced, our method does not suffer from the computational
cost of imposing the boundary values as described in [4] or similar approaches.
In the following the hierarchical grid coarsening will be explained in detail, followed
by the appropriate extension of Washio’s and Oosterlee’s error analysis and some numerical
experiments.
2. Solution using hierarchical grid coarsening. Assuming that we are interested in
the solution of
u
v *"w$&C'D' *')'+"-,k
zy){ S SW| /0C} x
a a , where supp inside of x
regular grid with mesh-width 3 and
. In order to do so, we are discretizing
1 .x using a
using the standard 7-point discretization
2.1. Extension
grid. The original grid is extended with the help of a grid extenz N s ofin the
sion rate p
the following way: The grid on the finest level is defined to be the
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72
M. BOLTEN
F IG . 2.1. Coarsened grid in 2D. Highlighted is the original fine grid, in which the solution is of interest.
discretization of domain x
a
with grid-width 3 , where
a
2
{
€
W
€
a aZ x a~
s s
€ av p 3 a~ 3 X
So x is just an extension of the original domain x . The domain is then extended and the
a
grid is coarsened up to a level ‚ max as
R{ € ƒ € ƒZ
xCƒ ~
s s
€ ƒ  p ƒ
3 ƒ ~ s…„ ƒ‡† aJˆ 3 X
The additional parameters € ƒ are introduced in order to enable the extended grids to have
common grid points with the fine grids. Furthermore we define the set of grid points ‰0ƒ on
level ‚ to be
‰ ƒ ~ Š8C x ƒ ' ‹ 3 ƒ = = >7? X
R
An example of how a coarsened grid might look like in 2-D can be found in Fig. 2.1.
2.2. Setting boundary conditions. At the boundary of xŒƒ max Dirichlet boundary conditions derived from the continous problem are imposed, i.e.,
+A WP N Q‹Ž 'D' I‹{6  ')' SB IA‘’“•” x ƒ X

max
2.3. Conservative discretization at the interface. To solve the problem on the composite grid
‰ ~ ‰ Aa – ‰ S ;
– —Y—<—:– ‰ ƒ
max
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HIERARCHICAL GRID COARSENING
73
œžŒ¦G¥+Ÿ¡ <¢ œžšŸ¡ <¢£œž*¤’¥+Ÿ6 <¢
˜š™
˜š§
˜+›
F IG . 2.2. Conservative discretization at the interface in 2D.
a discretization has to be chosen at the interfaces between x ƒ and x ƒ©¨
a<ª x ƒ . In order to obtain
good approximation properties, it is vital to require this discretization to be conservative. A
simple conservative discretization is the finite volume discretization using cubes around the
grid points.
As an example consider the two-dimensional discretization using finite volumes at the
refinement boundary depicted in Fig. 2.2 (the extension to 3-D is straightforward):
If we integrate the Poisson equation in a volume « and apply Gauß’s divergence theorem,
we get
Ž  9 Ž 6  ¬
¬
­¯® 6²³ µ  ²¶ Ž 6  X
—!´
 ¬±°
¬
For the rectangular elements we have to deal with, this equation can be approximated by the
sum of the fluxes times the length of the corresponding sides on the left hand side and by
the value at the center times the area of the element on the right hand side. The fluxes are
approximated using finite differences.
S
For square elements
this coincides with the standard 5-point approximation of the Laplace
operator times 3 , but we also have to deal with the elements on the boundary. The fine elements which have only one coarse neighbour are easy to treat, as the flux can directly be
approximated by the finite difference. The flux at the border of two coarse elements of a fine
element is the linear interpolation of the two neighbouring fluxes. So the flux ·*¸ in Fig. 2.2
is just set to
· ¸ sN · ƒ T · ¹ X
2.4. Solution of the resulting system of linear equations. Any solver can be used to
solve the resulting linear system, but a multigrid method is particularly well suited to solve
this problem, as it directly benefits from the structure of the composite grid ‰ . On each level
of the multigrid cycle only the grid points contained in the corresponding region are treated,
i.e., at level ‚ only the points in ‰ ƒ have to be taken into account. The information from the
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74
M. BOLTEN
whole problem is being transfered to this level by interpolation and through smoothing, as
the smoother uses information from xŒƒ©¨ when smoothing the boundary points of xŒƒ . The
a
resulting method is based on McCormick’s FAC (see [5, 6, 7]) and is given by:
Algorithm 1 FAC method for the solution of the resulting linear system.
º »4¼ smooth½ º» {7Á#1³Â g
¾
»4¼%¿À»
º »
¾ » ¨ ¼%Ã » » ¨ a ¾ »
»
a
º » ¨ ¼Åįú Æ » » ¨ a º »
a
¾ » ¨¨ a T Á
¿È» ¨ ¼ÅÄ ¿À» a a
¨
É» ¨ ¼ Á »† ¨ a ¿Àa » ¨ {
a à »» É » a a
É » ¼w
¨
º »4¼%º» ¨ T a É» a
º »4¼ smooth½cÊ º» º »¨
º »¨ a
» ¨ º» a ¨
a
º »¨
a
x
x
a
Ȭ
x »
» ¨ aAÇ x »
º» a<¨ ª x » ¨ x »
º» ¨ a x » ¨ a Ç x »
a
aª
This is a slight modification of the work of Washio and Oosterlee [9], who used Brandt’s
MLAT (cf. [1, 2]) and incorporated the conservative discretization at the boundary through a
conservative interpolation there.
2.5. Summary. The proposed modified method calculates the
discrete solution of the
Poisson equation inside of the unit square due to a right hand side , where the support of
is a subset of the unit square. The method
summarized as follows:
Ëcan
¡$& s be and
1. Choose grid extension rate p
maximum number of refinement steps
‚ max .
N s XYX<X ‚ , discretizing the appropriate domains
2. Generate grid hierarchy ‰Ìƒ ‚
max
xCƒ with mesh-widths 3!ƒ .
3. Set boundary values on the boundary points of the coarsest grid ‰0ƒ max to the values
derived from the continous problem.
4. Determine the composite grid ‰ containing all grid points from all levels, using a
conservative discretization scheme for the Laplace operator.
5. Solve the resulting system of linear equations.
3. Error analysis. For the error analysis, we start with a formal definition of the discrete
and the cell-averaged Green’s function.
“1 be a discretization of the Laplace
D EFINITION 3.18#(Discrete
Green’s
function).
Let
7
'
9
v
>
?
:
K
!
1
HcI be defined as
3= =
operator on the grid
and let
K<1&6HJI Ä $ N Í9±I
X
~
otherwise
Then the discrete Green’s function is defined by
G1 F B1 6HJI ÎK<1!HJI G1 is w.r.t. the first argument , only. ~
where
HcI
D EFINITION 3.2 (Cell-averaged
Green’s function). Let F
be the Green’s function
of the Laplace operator and let xŒÏ be defined as the cube with volume 3 centered at ,
i.e.,
I ÐÐ ')' v{;I#'D' ÑÓÒ 3 X
xCÏ ~ Ä 5
ÐÐ
sMÔ
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HIERARCHICAL GRID COARSENING
The cell-averaged Green’s function
ÕF
is given by
ÕF HJI #Ö3 N Ž F = J I  = X
B×
As we chose a conservative discretization, Green’s identity holds for the discrete case as
well. Thus we obtain
Ž y / 1ÙØ {±¡ 1 0 Ø |  9  ® y 1WØ {Ú 1 0 Ø | — Û ´  ² X

 °
°
1
Therefore, for the discrete Green’s function F , it holds true that

Ž F 1 6HJIYy©{¶2Ü 1 I |  I;
Ü 1 H{ ® y F 1 Hc²³ 1 Ü 1 6²³J{± 1 F 1 Hc²³JÝÜ 1 6²³ | Û  ² X
—´

°
°
(3.1)
With this observation, we are now ready to provide an error analysis for our modification.
T HEOREM 3.3. Using the described grid coarsening strategy up to an arbitrary level
S continuum problem at the
while imposing Dirichlet boundary conditions derived from the
boundary of the coarsest grid, using a grid extension rate p
s
 b and under the assumption
that
R
à a ތ')' S Z F 1 HJÞH(ÒßPWN Q ')' v{;N ތ')' ST ')' v{;
the error á
HcÞ , defined as
S
' ÕF 6HJÞHH{ F 1B6HJÞHY'©
á HcÞ ~ Ë
is of order 3 for all
‰!â .
Proof. We start by applying the discrete version of Green’s identity, inserting
into (3.1) and get
Ð
á HcÞ# ÐÐ Ž F 1 6HJIYy©{¶ 1 ÕF IAcÞA{ F 1 IAcÞc | 
ÐÐ !ã
1 Hݲ: 1 ÕF 6²@JÞH{ F 1 6 ²@JÞHJH{ 1 F 1 6Hc²³
®
 Ìã F
°
°
ä
max
max
á 6HJÞH
I T
ÕF 6²@JÞH { F 1 6²@JÞHJÀå — Û ´ 
² ÐÐ
ÐÐ
Ð
S
For an estimate of the first integral, we make use of the excellent work of Washio and Oosterlee [9]. It is well known that the accuracy of the solution on the finest grid is of order 3 .
So the error due to the integration over the finest grid is bounded by
' á:â 6HJÞHY'…Ò ³â 3 S X
à
Washio and Oosterlee showed that the error á due to the region outside the finest grid, but
a
not including the non-cubic-cells is bounded by
S
' á HcÞ<'…Ò
p{ { S N 3
a
à a N s mop  Ï  æV
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76
M. BOLTEN
and that the error á
S
due to the non-cubic cells is bounded by
S
' á SW6HJÞHY'…Ò S { N S
3 à N s mop  Ï W çÏ
 Ï and  æ are the minimum distances from the boundary of the finest
where
Þ , respectively.
/èMégrid
$B N of s
The proof depends on the fact that there exist constants »
à
such that
YX and
XYX ,
' ë» Ì† ê ìê F 6IAJÞHY'BÒ ' I{à Þ(» ' » < 6ítÒkè&
¨ a
I
Þ
where ë and ì act on and , respectively. For further details we refer to [9]. Overall,
the first integral can be bounded as follows:
ÐÐ
S
1
1
1
H
c
I
)
y
¶
{
*
I
J
H
Þ
{
6
A
I
J
H
Þ
J
I
|
 ÐÐ Ò :á â T á a T á S é]“ 3 X
F
ÕF
F
ÐÐ
Ð
Ð
S
It remains to show, that the second integral is bounded by an order 3 term as well,
S
i.e., that the setting of the boundary
values to the ones derived from the continuous problem
induces an error of order 3 . Let therefore  be the minimum distance of a point of the
original
to the boundary of the domain discretized using the coarsest grid. As both,
and Þ domain
are inside of the original domain, we can bound the second integral,
ÐÐ
ÐÐ Ž
ÐÐ Ìã
ÐÐ ®
ÐÐ 
Ð Ìã
max
F 1!Hc²³ Ì1 ÕF 6²@JÞH { F 1!6²@JÞHJ{ 1 F 1B6Hc²³ ÕF 6²@JÞH{ F 1!6²@JÞHJ å
°
°
ä
Ò p ƒ î2 ïoð Ð F 1Ì6Hc²³ 1B ÕF 6²@JÞH{ F 1B¡²WcÞc Û Ð T
ñóò !ã ÐÐ
— ´ ÐÐ
ä Ð 1 1 ° Hc²³ Û 6²@JÞH{ 1 ¡²WcÞc Ð å
F
ÐÐ F S
ÐÐ
— ´ ÕF S
S
°
Ð
Ð
Ð
Ð
Ð
3
n
3
n
3
Ò p ƒ ôBõ ÐÐ PWN Q N ÐÐ T Ð à a ƒ Ð Ð à a ƒ Ð T õ ÐÐ P@N Q N S ÐÐ T Ð à a ƒ ÐÐ ÐÐ à a 3
ÐÐ  ÐÐ ÐÐ  ÐÐ ÐÐ Wç ÐÐ
ÐÐ  ÐÐ ÐÐ @ç ÐÐ ÐÐ 
S Ð
SÐ ö Ð
Ð
S
ÐS
Ðö Ð
p ƒ ô ÐÐ P@N Q n à a 3 ƒ ÐÐ T ÐÐ n à a 3 çƒ ÐÐ T ÐÐ PWN Q n à a 3 ƒ ÐÐ T ÐÐ n à a 3 çƒ ÐÐ X
ÐÐ
 V ÐÐ ÐÐ +ø ÐÐ ÐÐ
+ø ÐÐ ÐÐ +ù ÐÐ ÷
Ð
Ð Ð
Ð Ð
Ð Ð
Ð
S Obviously, for p  s b we can bound  by
ƒ {
ƒ
 p s N  pP
Û  ² ÐÐÐ
— ´ Ð
Ð
max
max
max
max
max
max
and for 3
ƒ
max
max
max
max
max
max
we have
3ƒ
max
sš„ ƒ‡† Ja ˆ 3 sš„ ƒ‡† aJˆ 3 X
a
max
S
Ð
ƒ ÐÐÐ
Ð÷
max
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HIERARCHICAL GRID COARSENING
0.8
0.005
0.7
0
0.6
-0.005
0.5
-0.01
0.4
-0.015
0.3
-0.02
0.2
0.1
-0.025
1
1
0.8
0.6
0
0.2
0.4
0.4
0.6
0.8
0.6
0
0.2
0.2
0.8
0.4
0.4
0.6
0.8
1 0
0.2
1 0
F IG . 4.1. A cut through the computed solution of the test case (left) and its point-wise error (right).
So we get
p ƒ
max
ô ÐÐÐ PWN Q n à a 3 ƒ
V
ÐÐ
Ò p ƒ
S
ÐÐ
ÐÐ T
max
Ð
ô ÐÐÐ PWN Q
ÐÐ
ÐÐ N
ÐÐ PWQ
S
SÐ This is order 3 for pÿq±s b .
max
S
ÐÐ n 3 çƒ
ÐÐ à a +ø
Ð $+ú
n sàa3
p V
N sWsWýWý à
p ø
S
ÐÐ N n a 3 ƒ
ÐÐ T ÐÐ P@Q à @ ø
S Ð Ð S ƒ‡†
s „ aJˆ ÐÐ T
Ð
p
S û VBü S ƒ‡† ÐÐ
s „ a[ˆ ÐÐ
a3
ÐÐ
û p øü
Ð
max
ÐÐ
ÐÐ
ÐÐ T
max
Ð
ÐÐ N
ÐÐ
T
Ð
S
ÐÐ n 3 çƒ
ÐÐ à a @ù
Ð S
sWsÙý@ý à a 3 ç
p ø
S
ÐÐ P@þ N lWs 3
ÐÐ p ù à a
Ð
max
ÐÐ
ÐÐ
Ð÷
s ç „ ƒ‡† a[ˆ
û p øü
ç s ç „ ƒ†
û p ùü
ÐÐ
Ð
ÐÐ T
aJˆ ÐÐ
ÐÐ X
Ð÷
4. Numerical tests. The method has been implemented in C and tested to check the
S a machine with an 1.7 GHz Power4+
theoretical results. The performance was measured on
d
CPU. The grid extension rate p was set to N X q s b and for practical reasons € has been
chosen as
ã
€ ~ s Ê „
ˆ X
We used a cubic B-spline at the center of our domain of interest as the right hand side.
So the exact solution
to the problem is known analytically.
In Fig. 4.1 the computed solution of our test case and its error are shown. We calculated
the solution of the Poisson equation in free space for to a point-symmetric
hand side,
ú right
which can be described by a B-spline and which has unit volume on a N
grid. The error
behaves as expected: It is very similar to the solution of the problem with explicitly set
Dirichlet boundary conditions on the boundary of the computational box, but the error is not
exactly zero on the boundary. Table 4.1 gives the norm of the
, error and timings for different
grid sizes. One sees that the method scales linearly and the -norm of the error decreases as
expected.
In order to show that the number of refinement steps makes no difference with respect
to the method’s accuracy, we have tested the method using a different number of refinement
steps. The results of these tests can be found in Table 4.2. The norm of the error does
not change very much when the number of the refinement steps is varied, but of course the
timings differ a lot, as the number of boundary points is dramatically reduced and the size of
the linear system on the coarsest level as well.
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M. BOLTEN
TABLE 4.1
Error and timings for different various sizes. The -norm of the error decreases as predicted and the method
scales linearly with the number of grid points.
‰!ú â
N
@nd n l
Nsþ
3
NmNd
N mon+d Ps
Nm
N m N sÙý
refinements
ý
NWP N
Nú
N
s
l
n
N
'D' º {
X $+NWúN $WP$ N $
X ý sN
X sWl N n N n
X NWN sWl@lÙn
º D' ' Ñ S
$†
— NN $ † —N$†
—N$†ç
—
'D' º {
s XN d
N Xý N
N X lÙýþ
N Xn
º ')' S m $
$@s ý@lÙn+s@ll — NN $
$ þ N@N — N $
s ú n d —— N $
‰!† â
† Vø
†ù
†
time
1.66 s
12.84 s
104.61 s
909.64 s
TABLE 4.2
Timings and error norms for a -problem with and various refinements. The error of the method
is only marginally affected by the number of refinement steps.
refinements
Pn
s
dl
ýþ
ú
N$
NWN
Ns
N Pn
N
N dl
Nú
N
N ýþ
N
s$
‰!d ƒ
d ll
nú
n@n
n@n
s úN
Nú
N
N nþ
þ
þ
þ
þ
þ
þ
þ
þ
þ
max
')' º {
$
l X $ ý þ P N þÙP
l X $ ý@dWdl P sÙý
l X $ d ýWn
l X $šú n@þ sWýWPý
l X $ d ú l@l $
l X $šúo$ s@s d
l X $ P n+Ps
l X $ ý P $N ý
l X $šýú P s N
l X $ Pý P s N
l X $ ý $l ú N
l X $ ýWýþ ýþ
l X $ ýþ $ ý $ l
l X $ þ ý d $l
l X $ þ N Ps þ
l X $ þ N d ý$
l X $ þ N d@d n$
l X$ þ N d
l X N ýWý
º D' ' Ñ
$†
— NN $ † —N$†
—N$†
—N$†
—N$†
—N$†
—N$†
—N$†
—N$†
—N$†
—N$†
—N$†
—N$†
—N$†
—N$†
—N$†
—N$†
—N$†
—
'D' º { º ')' S m s X $ sÙn+ú@sWú s@ds — N $$
N X ý@þ l ú nú þ — N $
N X P s $ þl d P — N $
N Xý P — N $
N X ý N l@l N — N $
N Xý N l N l N — N $
N X ý NWN ý@ú l@s — N $
N X ý N s P s@s — N $
N X ý N $s ý@ý — N $
N X ý N Pý@s@l — N $
N X ý N s ú dl@l — N $
N X ý N Pn n — N $
N X ý N P+þlWsWn — N $
N X ý N sWýú — N $
N X ý N l N P@n P — N $
N X ý N l@s þ ú — N $
N X ý N l@s P — N $
N X ý N lWn@s ú — N $
N X ý N lWnWn — N
‰† â
† ø
ø
† ø
† ø
† ø
† ø
† ø
† ø
† ø
† ø
† ø
† ø
† ø
† ø
† ø
† ø
† ø
† ø
† ø
We ran the same test using the original method presented in [9], thus not setting the
boundary values to the values of the continuous problem. We used FAC instead of MLAT for
solving the linear system. The results in Table 4.3 and Fig. 4.2 show that this method behaves
as expected: Increasing the number of grid refinements increases the accuracy of the method
up to the same level than our modification.
Further we tested our modified method in a particle simulation code. Therefore the total
electrostatic energy of a DNA fragment including counter ions consisting of 1316 atoms was
calculated using different mesh sizes. Similarly to the example above, the point charges were
replaced by cubic B-spline charge densities. This has been corrected afterwards by a near
field correction. The results of this tests can be found in Table 4.4. One clearly sees, that the
relative error of the energy is divided by four as the grid size is doubled, as expected.
5. Conclusion. We have presented a method that uses hierarchically coarsened and extended grids in order to compute the solution of the Poisson equation in free space. The
method is similar to the method proposed by Washio and Oosterlee in [9], but the refinement
process is stopped at some level and the Dirichlet boundary conditions derived from the con-
ETNA
Kent State University
etna@mcs.kent.edu
79
HIERARCHICAL GRID COARSENING
TABLE 4.3
Timings and error norms for a -problem with and various refinements using the method of
Washio and Oosterlee. The error of the method heavily depends on number of refinement steps, reaching the same
accuracy as the modified method.
refinements
‰Ìd ƒ
d ll
nú
nWn
nWn
s úN
Nú
N
N nþ
þ
þ
þ
þ
þ
þ
þ
þ
þ
s
Pn
dl
ú
ýþ
N$
NWN
Ns
N Pn
N
N dl
Nú
N
N ýþ
N
s$
10
max
'D' º { º ')' Ñ S
d
n X ú Wl n@$l þ dú — N $$ †† S
N X ý@Pn þs — N $ S
N X þWlÙýþ P@ý P s — N $ †† Pý XX d n lN n N ýs — NN $ † n X þú d þs $ P d —— N $$ †† Ps XX $ sN P n+N lWlÙns — NN $ † P X n ú l N dWd — N $ † P X lWl P $ d P — N $ † P X ý@s N ý+sWs — N $ † P Xþ l d ú s ú — N $ † l X $$ s Pú þs@ds þ N —— N $$ †† l X $+lú P P — N $ † l X $ ý n$ — N $ † l X $ ý@n@ú P+sWþý — N $ † l X$ ý þ d $ ý — N $ † l X $ ýþ $ d@d ý — N $ † lX
s —N
−3
'D' º { º 'D' S m þ $ dWd $
þ N XX l úN n+l lWsÙn — NN $
$ ú — $
Pý XX lú@ú ý+þ@þsWýl P — NN $
s X n@þ $ýWn+s P l —— N $$
N X þ l@lú P NWN — N $
d N XX $šN ú ns P $ nl — NN $
P X s NWN ú l d — N $
n X n PP d sú þd l —— N $$
s X þs $ ý ý — N $
N X sWý@PsÙý — N $
N X ý+$s þ N ý$ úN — N $
N Xý $ þ ú N — N $
N X ý þ ýWú ý — N $
N X ý NWN s — N $
N X ý N Pn+l þÙN Ps — N $
N Xý N P n d — N $
N Xý N ý n — N
−1
10
original method
modified method
original method
modified method
−4
||e||2
||e||∞
10
‰† â
ç
†V
†V
†V
†V
†V
†V
†ø
†ø
†ø
†ø
†ø
†ø
†ø
†ø
†ø
†ø
†ø
†ø
10
−5
10
−6
−2
10
−3
5
10
15
# grid refinements
10
20
5
10
15
# grid refinements
20
F IG . 4.2. Behavior of the error of the original method and of our modification. Using the original method
both, the error in the -norm and in the -norm, depend heavily on the number of grid refinements. The accuracy
converges to the accuracy of our modification, that is almost independent of the number of refinements.
TABLE 4.4
Relative error of the total electrostatic energy of a 1316 atom DNA fragment calculated using a particle simulation code with the help of our modified method.
‰!ú â
N @nd n l
Nsþ
3d
NmN
N mon+d Ps
Nm
N m N sWý
ý
l
s
N
' é{!
X ú $n ú n úú@þú
X nú n+d þs $
X núþN
X sÙý n@n
'm '
$
— NN $
—N$
—N$
—
'
†ç
†ç
†V
†
V
ETNA
Kent State University
etna@mcs.kent.edu
80
M. BOLTEN
tinous problem are imposed on that level. This is a major difference to their method, which
in principle refines the grid infinitely many times.
We extended Washio’s and Oosterlee’s error analysis and we have shown that the introduction of the boundary conditions at the coarsest level does not influence the order of the
method’s accuracy. Therefore the modification enables the use of hierarchical grid coarsening without altering the accuracy by choosing a different number of refinement steps at very
small costs.
We implemented the modified method. The numerical experiments show that the error
behaves as expected and that the stopping at arbitrary levels does indeed not significantly
influence the behaviour of the computed solution. Thus, the proposed method is suitable to
compute the solution of Poisson’s equation in free space using a multigrid method.
In the future, we plan to further optimize our implementation and to add higher-order
conservative discretization schemes to both our analysis and our implementation. Additionally, we will work on the parallelization of our method, so that we can use it in large scale
molecular dynamics simulations.
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, Multi-level adaptive solutions to boundary value-problems, Math. Comp., 31 (1977), pp. 333–390.
[2]
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[4] R. H. B URKHART , Asymptotic expansion of the free-space Green’s function for the discrete 3-d Poisson
equation, SIAM J. Sci. Comput., 18 (1997), pp. 1142–1162.
[5] S. M C C ORMICK , Multilevel Adaptive Methods for Partial Differential Equations, SIAM, Philadelphia, 1989.
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